Can somebody give me a hint or help me to solve this problem.
Let V be a p×q matrix with orthonormal columns (p > q), and $M = I−2VV^T$ , with I being the p × p identity matrix. The matrix M can be viewed as a “generalization” of a Householder matrix.
(a) Show that $||M||^2 = 1$. (b) Compute the Frobenius norm of M
(a) Compute $Mv$ with $v$ being a column of $V$. Compute $Mx$ for a vector $x$ orthogonal to all columns of $V$ ($x^TV=0$). Take a suitable orthonormal basis of the space to conclude.
(b) $\|M\|_F^2 = tr(M^TM)$ and take advantage of $V^TV = I_q$.